Rearrangement Invariant Banach Function Spaces Research Articles

Two results on rearrangement invariant spaces on an infinite interval

We construct (Theorem 2) an example of a nonseparable rearrangement invariant Banach function space X on (0, ∞) that satisfies the following “truncated” shift continuity condition (S): ∥ xχ ( n, n + 1) ∥ → 0 as n → ∞ for each x ϵ X. On the other hand (Theorem 1) condition (S) implies almost separability, that is, that the restriction of an arbitrary space (satisfying (S)) to any subset of finite measure is separable.

The Marcinkievicz Interpolation Theorem for Rearrangement-Invariant Function Spaces and Applications

The interpolation theorem of J. Marcinkievicz [17] states that any sublinear operator T which is simultaneously of weak types ( p_1, q_1 ) and ( p_2, q_2 ) is also a bounded operator from the Lebesgue space L_p(0, l), 0 < l \leq \infty , into itself, provided p_2 < p < p_1 . The aim of this paper is to generalize this theorem to the setting of rearrangement-invariant Banach function spaces, and thus to render the theorem available to a much larger range of applications.